QUA 



/ JL", the fluxion of the area A B R is, in 



b x x 



this case, equal to ; and conse- 

 quently the fluent thereof, or the area 



b x* 

 itself, =-^ which, therefore, when x = 



a, and B R coincides with H M, will be- 

 ab AHxHM 



come -q- o == area 



whole triangle A H M : as is also demon- 

 strable from the principles of common 

 geometry. 



Again, let the curve A R M H, (fig. 7.) 

 whose area yon would find, be the com- 

 mon parabola ; in which case, if A B = a-, 

 and B R = y, and the parameter = a ; 



we shall have y* =- ax, and^y =3 a 2"x2: 



and therefore, u 

 whence u =3"X a 



2. 1 J 

 -sosa* x ^ = 



|#.r s =.|xAB X B R. Hence a pa- 

 rabola is two thirds of a rectangle of the 

 same base and altitude. 



The same conclusion might have been 

 found more easily in terms ofy for x = 



v* 



2, and x = 



2 y ii 



- 



} = -; whence u = 



n 3 a ~~ 



R, as before. 



To determine the area of the hyper- 

 bolic curve A M R B, (fig. 8) whose 

 equation is x m y n = am+n whence we 



m+n m+n 



have y = on = a 

 xn 



and 



m+n 



therefore u (= y x ) = a n X 

 m+n m 



an i__ n 



VX >y 



whose fluent is u = 

 m+n nm 



; which, when x = 0, will 



71 m 



also be =0, if n be greater than m; 

 therefore the fluent requires no correc- 

 tion in this case ; the area, A M R B, in- 

 cluded between the asymptote, A M, and 

 the ordinate B R, being truly defined by 

 m+n nm 



n an x x 



~ , as above. But if n be 



n m 



less than m, then the fluent, when x = 0, 



QUA 



will be infinite, because the index n *" 



n 



being negative, becomes a divisor to 

 n am+n ; whence the area, A M R B, will 

 also be infinite. 



But here, the area, B R H, compre- 

 hended between the ordinate, the curve, 

 and the part, B H, of the asymptote, is 

 finite, and will be truly expressed by 



, the same quantity with 



m n 



its signs changed ; for the fluxion of the 



m+n m 



part A M R B, being an x xn , 



that of its supplement B R H must con- 



m+n m 



sequently be an x xn , whereof 

 m+n i-m m+n nm 



an x Xn an x xn 



m n 



if a n 



the fluent is ----- 



; and consequently u 



= the area, B R H, which wants no cor- 

 rection ; because when x is infinite and 

 the area B R H = 0, the said fluent will 

 also entirely vanish ; since the value of 

 mn m+n 



} which is a divisor to a 3 is then 

 infinite. 



For further examples see Simpson's 

 Fluxions, vol i. sect. vii. 



QUADRATURE, in astronomy, that aspect 

 of the moon when she is 90 degrees dis- 

 tant from the sun ; or when she is in a 

 middle point of her orbit, between the 

 points of conjunction and opposition, 

 namely, in the first and third quarters. 



QUADRATURE lines, are two lines placed 

 on Gunter's sector .-they are marked with 

 Q. and 5, 6, 7, 8, 9, 10 ; of which Q. signi- 

 fies the side of the square, and the other 

 figures the sides of polygons of 5, 6, 7, 

 &c. sides. S, on the same instrument, 

 stands for the semi-diameter of a circle, 

 and 90 for a line equal to ninety degrees 

 in circumference. 



QUADRILATERAL, in geometry, a 

 figure whose perimeter consists of four 

 right lines making four angles ; whence 

 it is also called a quadrangular figure. 

 The quadrilateral figures are either a pa- 

 rallelogram, trapezium, rectangle, square, 

 rhombus, or rhomboides. 



QUADRUPEDS, in zoology, a class of 

 land animals, with hairy bodies, and four 

 limbs or legs proceeding from the trunk 

 of their bodies : add to this, that the fe- 

 males of this class are viviparous, or bring 

 forth their young alive, and nourish them 

 with milk from their teats. This class, 



